The co-ordinates of the end points of the latus rectum of a parabola are (3,6) and (-5,6). The co-ordinates of its focus are :

Options

A. (0,0)
B. (1,1)
C. (2,2)
D. None of these.

Answer

To find the coordinates of the focus of the parabola given the endpoints of its latus rectum, we can follow these steps:

### Step 1: Identify the endpoints of the latus rectum
The endpoints of the latus rectum are given as \( A(3, 6) \) and \( B(-5, 6) \).

### Step 2: Use the midpoint formula
The focus of the parabola lies at the midpoint of the latus rectum. The midpoint \( F(x, y) \) can be calculated using the midpoint formula:
\[ F(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \).

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of points \( A(3, 6) \) and \( B(-5, 6) \) into the midpoint formula:
\[ F(x, y) = \left( \frac{3 + (-5)}{2}, \frac{6 + 6}{2} \right) \]

### Step 4: Simplify the calculations
Calculating the x-coordinate:
\[ x = \frac{3 - 5}{2} = \frac{-2}{2} = -1 \]
Calculating the y-coordinate:
\[ y = \frac{6 + 6}{2} = \frac{12}{2} = 6 \]

### Step 5: Write the final coordinates of the focus
Thus, the coordinates of the focus \( F \) are:
\[ F(-1, 6) \]

### Final Answer:
The coordinates of the focus of the parabola are \( (-1, 6) \).

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